PRONEU 1317 S0301-0082(13)00145-7 10.1016/j.pneurobio.2013.12.005 Fig. 1 Comparison of the default mode network (DMN) in rats, monkeys and humans. The regions composing the DMN exhibit correlated neuronal activity during rest. Adapted fromLu et al. (2012). Fig. 2 Consistent resting-state networks across healthy subjects detected with fMRI. 10 distinct patterns with potential functional relevance were detected using tensor probabilistic independent component analysis, consisting of regions known to be involved in motor function, visual processing, executive functioning, auditory processing, memory, and the default-mode network, each with BOLD signal changes up to 3%. All functional images have been co-registered into a standard structural MRI template (MNI). Adapted fromDamoiseaux et al. (2006). Fig. 3 Amplitude fluctuations of band-passed oscillations at different frequency bands. The amplitude envelope (red) of oscillations filtered in a restricted frequency range (black) are proportional to the squared root of the power. Recent EEG and MEG studies point to a relationship between these envelope fluctuations and the BOLD signal fluctuations observed during rest. However, the frequency range (or carrier frequency) that best captures BOLD fluctuations remains under debate. Fig. 4 Comparison of brain networks obtained using ICA independently on MEG and fMRI data. (A) DMN; (B) left lateral frontoparietal network; (C) right lateral frontoparietal network; (D) sensorimotor network; (E) medial parietal regions; (F) visual network; (G) frontal lobes including anterior cingulate cortex; (H) cerebellum. All networks were identified using beta-band power fluctuations, except the DMN, which was detected using alpha-band power fluctuations. Adapted fromBrookes et al. (2011). Fig. 5 Power envelope correlations between orthogonalized spontaneous signals from homologous early sensory areas. (a) Inter-hemispheric correlations between auditory (b), somatosensory (c) and visual (d) cortices, as a function of the carrier frequency. Correlations are increased in the alpha and beta frequency ranges (8-30Hz). Adapted fromHipp et al. (2012). Fig. 6 Exploring the structure-function relationship using computational models. Diagram illustrating how computational models can serve to explore the relationship between anatomical structural networks and resting functional networks. The dashed line (--) indicates the feedback of the model's performance, that is necessary to tune the parameters of the model. Fig. 7 The Macaque connectivity. The coupling weightsCnpindicate the strength of connections (classified as weak (1), medium (2) or strong (3)) from regionpton(directed). The CoCoMac database, created by Rolf Kotter, represents one of the first attempts to obtain a large-scale anatomical connectome of the mammalian brain. Fig. 8 White matter fibres detected in vivo using Diffusion MRI. Images from the gallery of the Connectome Project by R. Buckner. Fig. 9 Anatomical network with 66 cortical regions. Anatomical connectome derived byHagmann et al. (2008)using DSI averaged over 5 healthy subjects. Top-left: Parcellation scheme dividing each hemisphere into 33 anatomically segregated regions (adapted fromHagmann et al. (2008)). Top-middle: White matter tracts detected using DSI and tractography. Top-right: Schematic representation of the anatomical network, where regions are represented by red spheres placed at their centre of gravity and the link's thickness is proportional to the number of fibre tracts detected in each connection. Bottom-left: The coupling weights are proportional to the number of tracts detected. White colour means that no fibre connecting the two corresponding regions was detected. Weights were normalized so that 0=1s). The authors suggested that the low-frequency amplitude modulation of spontaneous rhythmic activity may be the source of correlations of low-frequency haemodynamic responses, generally interpreted as functional connectivity. This work appears to be the first demonstration of band-limited power correlations between the two hemispheres during rest. Almost a decade later, and following the electrophysiological studies indicating that spontaneous BOLD fluctuations could be driven by slow amplitude modulations of neural oscillations,Liu et al. (2010)characterized the power modulations of spontaneous MEG rhythms recorded from human subjects during wakeful rest (with eyes open and eyes closed) and light sleep. Through spectral, correlation and coherence analyses, they found that resting-state MEG rhythms demonstrated ultraslow (<0.1Hz) spontaneous power modulations that synchronized over a large spatial distance, especially between bilaterally homologous regions in opposite hemispheres. Their observations suggest that coherent power modulations of spontaneous rhythmic activity (especially in the beta-band) reflect the electrophysiological signature of the large-scale functional networks. A couple of months later de Pasquale and colleagues used a seed-based method to characterize the MEG signatures of two well-characterized RSNs: the dorsal attention and the default mode networks (de Pasquale et al., 2010). Taking into account the non-stationarity of MEG activity, they found that the band-limited power of the RSN seeds were only synchronized for restricted periods in time, resulting in a transient formation of RSNs. Their results indicate that RSNs manifest in MEG as synchronous modulations of band-limited power primarily within the theta, alpha, and beta bands, which correspond to rhythms slower than thegamma-frequency oscillations generally associated with the electro-physiological correlates of event-related BOLD responses. The first work to show that MEG can independently detect the spatial patterns of RSNs, in the same manner that has been demonstrated in fMRI, was recently performed by Brookes and colleagues (Brookes et al., 2011) As a first step, the MEG data was frequency filtered into bands of interest (delta,theta,alpha,beta, andgamma) and projected into source space using a beamformer spatial filter. Then, ICA was applied to the band-limited amplitude fluctuations, resulting in 8 RSNs with significant similarity to the RSNs derived independently using fMRI (seeFig. 4). Importantly, most resting-state networks were linked to beta-band amplitude fluctuations. This outcome confirms the neural basis of haemodynamic networks and demonstrates the potential of MEG as a tool for understanding the mechanisms that underlie RSNs. In a recent MEG study, Hipp and colleagues found that spontaneous oscillatory neuronal activity exhibits frequency-specific spatial correlation structure in the human brain (Hipp et al., 2012). To discount spurious correlation of signal power caused by linear signal leakage, a new analysis approach was developed in which the source estimates of spontaneous neuronal activity reconstructed from MEG are orthogonalized, discarding in this way in-phase relationships. They found power envelope correlations between homologous early sensory areas (seeFig. 5). Overall, correlation of power across cortical regions was strongest in the alpha to beta frequency range (8-32Hz) and correlation patterns depended on the underlying oscillation frequency. Furthermore, they identified frequency-specific hubs residing in the medial temporal lobe for the theta frequency range (4-6Hz), in lateral parietal areas for the alpha to beta frequency range (8-23Hz) and in sensorimotor areas for higher frequencies (32-45Hz). These results reinforce the idea that interactions in various large-scale cortical networks may be reflected in frequency-specific power correlations. MEG has a higher spatial resolution than EEG because of the higher sensitivity of MEG sensors and because magnetic fields are less distorted by the skull and scalp than electric fields (Hansen et al., 2010). Making the appropriate corrections to account for linear signal leakage (Brookes et al., 2012; Hipp et al., 2012) the spatial resolution of MEG can be as precise as 2mm for sources in the cortical surface. However, MEG detects only the tangential components of cortical activity (from the sulci) and the local magnetic fields originating in deep (subcortical) sources may be distorted, limiting spatial accuracy. As such, the analysis of resting-state MEG data has often been restricted to cortical areas. 2.4 Electrophysiological correlates of resting-state BOLD fluctuations All electrophysiological studies presented in the previous sections have aimed to investigate the neurophysiological counterpart of resting-state BOLD fluctuations. Overall, results indicate that resting-state BOLD fluctuations are driven by slowmodulations of neural activity, which are manifested by transient increases in the amplitude (or in the power) of fast oscillations in a certain frequency range. However, regarding the question whether there is a specific carrier frequency whose power modulations serve as the best candidate for the neuronal correlates of spontaneous BOLD signal fluctuations, there seems to be a debate. On one side, the intra-cortical recordings performed byLeopold et al. (2003),Nir et al. (2008)andShmuel and Leopold (2008)propose a direct relationship with the power of gamma-frequency oscillations. Although restricted to the recording site, their studies indicate that the BOLD signal is positively correlated with the local gamma LFP. To the extent that their findings can be generalized to other cortical areas, these findings indicate that BOLD functional connectivity between remote regions during rest can be linked to slow increases in neuronal activation levels happening in the gamma-band. Indeed, the elevated energy requirements of high-frequency neural oscillations represent a mechanistic link between the functional connectivity of brain regions and their respective metabolic demands (detected in the BOLD signal) (Lord et al., 2013). On the other hand, most of the studies using external recordings, such as EEG and MEG (Brookes et al., 2011; de Munck et al., 2007; de Pasquale et al., 2010; Difrancesco et al., 2008; Feige et al., 2005; Goncalves et al., 2006; Hipp et al., 2012; Laufs et al., 2003; Liu et al., 2010; Moosmann et al., 2003; Nikouline et al., 2001) have identified correlated fluctuations in the power (or the amplitude) of oscillations in the alpha and beta frequency ranges, revealing functional networks that closely match the ones obtained in fMRI studies. This relationship was investigated using simultaneous recordings of EEG-fMRI and results point to an inverse correlation between the BOLD signal and the power in the alpha band, indicating that these "idle rythms" may be associated with deactivation and decreased metabolic rate. Since each technique has intrinsic limitations and the results are not exclusive, both scenarios could coexist. Indeed, according to the results fromScheeringa et al. (2011), a negative correlation exists between the powers at high and low frequencies, indicating that the BOLD signal might be associated to interactions between gamma-band activity (related to increased metabolic rate) and alpha/beta-frequency oscillations (decreased metabolic rate). Furthermore, rhythms from different frequency bands lead to different correlation patterns, suggesting a complex interplay between multiple frequency bands (de Pasquale et al., 2010; Mantini et al., 2007) which in turn may reflect different aspects of neuronal processing (Palva and Palva, 2012; Womelsdorf et al., 2007). However, an accurate answer requires simultaneous recordings of both fMRI and electro-physiological signals across the whole brain, which is unavailable in the present days. Alternatively, one can explore the mechanism at the genesis of these slow electrophysiological power fluctuations and their relationship with the BOLD signal by means of computational models, which serve to test theoretical predictions (in Section4we explore these models in detail). 2.5 Altered resting-state activity in disease Over the last decade, a large number of studies have reported altered resting brain activity in a wide range of mental illnesses. These results not only illustrate the importance of balancing resting-state dynamics for an optimal cognitive function, but also provide insights to understand the intrinsic mechanisms leading to and potentially treating the diseased brain (Kringelbach et al., 2011). However, it is important to note that substantial methodological limitations arise when recording data from patients with psychological disorders due to higher motion and physiological artefacts. The correction of these artefacts likely reduces any significant BOLD signal and therefore the alterations reported in resting-state activity in disease must be discussed with caution. In Alzheimer's disease (AD), for example, functional connectivity during rest was found to deteriorate within all systems with the progression of the disease (Binnewijzend et al., 2012; Damoiseaux et al., 2012; Greicius et al., 2004; Zhou et al., 2010). Furthermore,Supekar et al. (2008)characterized resting-state functional networks using graph theory and found that functional brain networks in AD showed loss of small-world properties, characterized by a significantly lower clustering coefficient. Regarding schizophrenia, several studies have reported a widespread decrease in the functional connectivity of patients during rest (Bassett et al., 2012; Liang et al., 2006; Lynall et al., 2010; Skudlarski et al., 2010), supporting the hypothesis that schizophrenia may arise from the disrupted functional integration of segregated brain areas. Further analysis of resting-state activity in schizophrenia using graph theory indicate a subtle randomization of functional networks, with decreased small-world properties, lower clustering coefficients and less high-degree hubs (Liu et al., 2008; Lynall et al., 2010; Bassett et al., 2012). Resting-state alterations have been found in many other mental illnesses including dementia (Buckner et al., 2000; Rombouts et al., 2009), autism (Cornew et al., 2012; Cherkassky et al., 2006; Kennedy et al., 2006; Lai et al., 2010; Weng et al., 2010), mild cognitive impairment (Rombouts et al., 2005), multiple sclerosis (Bonavita et al., 2011; Faivre et al., 2012; Schoonheim et al., 2013) and major depression (Greicius et al., 2007; Sheline et al., 2009; Veer et al., 2010). Despite the strong evidence of disrupted resting-state activity in disease, the use of resting-state data for clinical purposes is still in progress since it requires consistency across studies before it can be used in a meaningful way at the single-patient level (Greicius, 2008; Fox and Greicius, 2010). In some cases, resting-state functional connectivity was found to correlate with cognitive performance (i.e.Lynall et al., 2010), which indicates that resting-state correlations may be closely related to the binding mechanisms that support the integration of information in the brain. Overall, more detailed investigations of the highly coherent functional and structural brain networks in health and disease have the potential not only to increase our understanding of fundamental brain function but of how best to modulate the balance. In particular, some of the problems found in neuropsychiatric disorders could inform future treatment, e.g. potentially with deep brain stimulation (Kringelbach et al., 2011). 3 Networks of the brain Mapping the brain is complex but is helped by the fact that the brain, in the same way as many biological, social and chemical systems, is a network composed by a large number of interconnected dynamical units. The map of neural connections in the brain has been called the Connectome (Hagmann, 2005; Sporns et al., 2005), a term inspired by the effort in mapping the human genetic code, or Genome. Connections in the brain may refer to anatomical links, such as synapses or fibre pathways (anatomical/structural connectivity), statistical dependencies such as correlation or coherence (functional connectivity) or to directed causal interactions (effective connectivity) between distinct units within a nervous system. Such units may correspond to individual neurons, neuronal populations, or anatomically segregated brain regions. Exploring the whole connectome at a cellular scale is highly complex and hardly tractable, since the human cerebral cortex contains at least 1010neurons linked by 1014synapses (Sporns, 2009). The only organism from which the full map of neuronal connections is known in its entirety is the one millimetre-long wormCaenorhabditis elegans, but it took over a decade to complete the identification of its 300 neurons and 7000 connections (White et al., 1986). In the context of resting-state activity, it is convenient to go beyond the cellular scale and focus essentially on the long-range white matter pathways connecting segregated brain areas. These long-range connections are particularly important because they strongly constrain how brain regions communicate, defining the wiring diagram over which spatially segregated information is integrated in the nervous system (Jirsa and Kelso, 2000; Sporns et al., 2000; Tononi et al., 1992, 1994). Spatially distributed and functionally specialized brain areas are continuously communicating and co-operating to respond to cognitive demands, perceive sensory stimuli and generate coordinated movement. Although the neuronal mechanisms under which cortical regions communicate remain unclear, the neuroanatomical network is believed to serve as the structural substrate upon which coordinated functional integration occurs (Bressler and Tognoli, 2006; Ghosh et al., 2008a; Jirsa and McIntosh, 2007; Knock et al., 2009; Sporns et al., 2000). Combining functional and structural imaging modalities has revealed that resting-state functional networks reflect, to some extent, the underlying structural connectivity (Damoiseaux and Greicius, 2009; Greicius et al., 2009; van den Heuvel and Hulshoff Pol, 2010; van den Heuvel et al., 2009a). However, the bond between structural and functional connectivity is not straightforward: although structural connectivity is a good predictor of functional connectivity - i.e. if there is a direct anatomical connection, there is likely a functional connection - the opposite is not necessarily true. Indeed, robust functional connectivity has been observed in the absence of a direct anatomical link (Koch et al., 2002). Recent studies suggest that resting-state functional connectivity also reflects polysynaptic anatomical pathways related tocerebro-cerebellar circuits (Buckner et al., 2011; Habas et al., 2009; Krienen and Buckner, 2009; O'Reilly et al., 2010). Indeed, it was found that spontaneous fluctuations in the motor cortex are functionally coupled to their topographically appropriate contra-lateral regions in the anterior lobe of the cerebellum (Buckner et al., 2011). Furthermore, patients with focal pontine strokes, were found to display selectively reduced functional connectivity between the motor cortex affected and the contra-lateral cerebellum, providing additional evidence that resting-state functional connectivity reflects polysynaptic anatomical pathways (Lu et al., 2011). Considering that coupled brain regions have a causal relationship and the connectivity is directed, the functional network architecture may be inferred using Granger Causality (GC) or Dynamic Causal Modelling (DCM) (Friston et al., 2003, 2013; Kasess et al., 2010; Liao et al., 2010; Penny et al., 2004; Stephan et al., 2008). These methods are principally used to test or predict dynamical responses to causal events (Marreiros et al., 2010). Recently,Friston et al. (2011)proposed an extension to conventional DCM to include endogenous or random fluctuations and allow in this way the application of DCM to resting-state BOLD time series. However, its application to resting-state data is still limited and so far restricted to the DMN (Di and Biswal, 2013; Li et al., 2012). While the origin of resting-state functional connectivity remains unclear, computational models of neural network dynamics (which we describe in detail in Section4) are valuable tools to investigate the role of the anatomical network in shaping functional connectivity during rest (Fig. 6) (Ritter et al., 2013). 3.1 Brain parcellation The first step to define brain networks is to define the nodes in the network. To do so, it is necessary to divide grey matter at the desired scale according to specific strategies for anatomical or functional partitioning. Several brain parcellation templates are available in the literature, ranging from the canonical classification into lobes (i.e. frontal, parietal, temporal and occipital lobes) to as much as several thousand regions of interest. Once the parcellation scheme is defined, different strategies can be used to define the links between nodes. One of the first and most widely known parcellation schemes of the human cortex was performed by the German anatomistBrodmann (1909)based on the cytoarchitectural organization of neurons. In 2002, a parcellation scheme was proposed by Tzourio-Mazoyer et al. with the intention of standardizing the anatomical labelling of brain regions in neuroimaging studies (Tzourio-Mazoyer et al., 2002). Automated anatomical labelling (AAL) was performed on a brain template consisting on a high-resolution structural MRI scan from a healthy male supplied by the Montreal Neurological Institute (MNI) (Collins et al., 1998). The cortical and subcortical grey matter of the standard MNI brain was divided into 90 brain regions (45 in each hemisphere). The AAL parcellation template is included in the Statistical Parametric Mapping (SPM) package (Friston et al., 1994) and is freely available to the neuroimaging community. To date, more than 2000 research articles have used the AAL parcellation template (based on citation index), including studies of resting-state functional connectivity (Achard and Bullmore, 2007; Achard et al., 2006; Bassett et al., 2012; Braun et al., 2012; Liu et al., 2008; Lynall et al., 2010; Salvador et al., 2005b; Sanz-Arigita et al., 2010; Supekar et al., 2008). Another parcellation scheme used in studies of large-scale brain connectivity is the one created byHagmann and colleagues (2007). Grey matter was divided into 66 cortical regions and then individually subdivided into small regions of interest (ROIs) resulting in 998 ROIs, each covering 1.5cm2of the cortical surface. Two labelled meshes (one mesh with the 66 regions and another with the 998 ROIs) were created on the average brain and were registered onto the brain of individual participants using Freesurfer (surfer.nmr.mgh.harvard.edu). Recently, the same group proposed a robust method for constructing normalized parcellations at different scales, iteratively regrouping the 998 ROIs into bigger ROIs, resulting in 5 scales of cortex parcellation, e.g. 66, 133, 241, 483 and 998 regions (Cammoun et al., 2012). The analysis of brain networks may also be performed at the voxel level, i.e. the smallest cubic element in a volumetric image. The exact size of a voxel will vary depending on the technology used. Typically, fMRI voxels represent a volume of 27mm3(a cube with 3mm length sides), which contain about 1 million neurons. Invan den Heuvel et al. (2009b), functional network properties were explored at the voxel scale (64mm3), resulting in graphs with ~9500 nodes. In another work from the same group, RSNs were defined by applying a voxel-based clustering method (van den Heuvel et al., 2009a). In large-scale models, however, the simulation of systems with several thousands of interconnected nodes is highly expensive from the computational perspective and low-dimension parcellation schemes have been typically used in models of large-scale resting-state activity. 3.2 Anatomical structural networks Historically, the mapping of white matter connections was performed post-mortem by histological dissection and staining, by degeneration methods or by axonal tracing. These classical techniques, however, were time-consuming and generally applied to restricted areas in the brain. In an attempt to assemble data from different white matter tracing studies of the macaque brain,Kotter (2004)created the online database for the "Collation of Connectivity on the Macaque brain" (CoCoMacwww.cocomac.org), allowing for continuous updating and refinement of such anatomical connection maps. From this database, Kotter and Wanke derived a realistic connectivity map of one hemisphere of the primate brain, proposing a coarse parcellation of the primate cerebral cortex into 38 regions, which reflected broad and rather uncontroversial divisions (Fig. 7) (Kotter and Wanke, 2005). This connectivity map has been used in models of large-scale resting-state brain activity (Deco et al., 2009; Ghosh et al., 2008a; Honey et al., 2007) revealing that the large-scale connectivity topology of cerebral cortex, together with time delays and in the presence of noise, defines a dynamic framework from which spontaneous patterns similar to the ones observed in the resting brain emerge. For human brains, however, the manual tracing of white matter pathways has not been performed. Some works have tried to infer human anatomical networks from cortical thickness/volume measurements obtained with structural MRI. This approach relies on the fact that cortical volume is strongly correlated between regions that are axonally connected (Lerch et al., 2006). In this way, a whole-brain network can be inferred from human MRI data by estimating, for each brain area, the cortical volume (or thickness) for a large sample of subjects and then computing the correlation matrix (Bassett et al., 2008; Chen et al., 2008; He et al., 2007). However, there is still no direct proof that correlations of grey matter volume over the whole brain across subjects are indicative of axonal connectivity. In recent years, a revolution inconnectomics(i.e. the science concerned with assembling and analyzing connectome data sets,Hagmann, 2005) has happened helped by the technological advancements in Diffusion MRI. This method permits the non-invasive detection of white matter fibre pathways using a specific MRI sequence that is sensitive to water anisotropy, i.e. the direction of water diffusion in a body. The detection of fibre tracts with diffusion MRI relies on the fact that water propagates along the orientation of the fibres because the myelin sheath provides a barrier perpendicular to the fibres. Diffusion tensor imaging (DTI) (Wedeen et al., 1995) estimates the main direction and strength of anisotropic diffusion in each voxel while diffusion spectrum imaging (DSI) explores the strength of anisotropy in all directions, allowing the crossing of multiple fibres in a single voxel (Wedeen et al., 2005). Once diffusion images are obtained, the detection of fibre tracts, or tractography, is obtained by constructing three-dimensional curves of maximal diffusion coherence using computational algorithms. Together with efficient computational tractography algorithms, this technique permits the rapid and almost automatic construction of comprehensive maps of brain connectivity. The Human Connectome Project (www.humanconnectomeproject.org) aims to provide an unparalleled compilation of neural connectivity data based on Diffusion MRI studies (Fig. 8). To define a low-resolution structural network using Diffusion MRI, the connection strengthCnp between two brain areasnandpis scaled by the number of white matter tracts detected between voxels in each area, which can range from 0 to as much as several thousand tracts per connection. The first mappings of human whole-brain anatomical connectivity using Diffusion MRI tractography were performed almost simultaneously byHagmann and colleagues (2007)andIturria-Medina and colleagues (2007). InHagmann et al. (2007), anatomical networks were composed by 500-4000 nodes and were derived from the brains of 2 healthy subjects. Subsequently, low-resolution parcellation was proposed, this time dividing the cortex into 66 regions (Fig. 9) (Hagmann et al., 2008). This anatomical network was used in the first large-scale model of human resting-state functional connectivity (Honey et al., 2009). Iturria-Medina et al. analyzed anatomical connectivity at lower resolution, first focusing on the connectivity between 71 grey matter structures, across 5 healthy subjects (Iturria-Medina et al., 2007) extended later to 20healthy participants and 90 cortical and subcortical regions according to the AAL parcellation scheme (Iturria-Medina et al., 2008). To date, the largest sample of healthy structural brain networks using DTI and the AAL parcellation was from 80 young adults (Gong et al., 2009). 3.3 Dynamic functional networks The simultaneous activation of spatially segregated functionally specialized brain regions defines a functional network, and these regions are said to be functionally connected. Unlike structural networks, functional networks are only active for a period in time, representing a transient brain state where different brain areas activate simultaneously, supposedly integrating segregated information (Tononi et al., 1998, 1994). Evokedfunctional networks can be easily captured by comparing a measure of brain activity during a particular task with baseline data sets. During rest, however, the definition of an activation paradigm becomes unfeasible. Still, even when no task-related process is triggering the activation of a certain functional network, several studies have reported the transient activation of distinct functional networks during rest. Functional connectivity can be inferred from resting-state recordings using different approaches, namely correlation measures, ICA, PCA, mutual information, covariance and coherence analysis. The classic and most widely used method to infer the strength of network interactions (or functional connections) consists in estimating the linear (Pearson) correlation coefficient between temporal signals (Bandettini et al., 1993; Biswal et al., 1995). If two regions activate and deactivate at the same time, there is likely a functional connection. The Pearson correlation coefficient between two seriesXandYof sizeNis given by the following equation:rho X , Y = ? n = 1 N ( X n - X ? ) ( Y n - Y ? ) ? n = 1 N ( X n - X ? ) 2 ? n = 1 N ( Y n - Y ? ) 2 Using correlation measures to investigate resting-state patterns, two levels of analysis are possible: (1) looking at the correlation between one specific region and the rest of the brain, i.e. seed-based correlation, or (2) exploring all possible functional connections by studying the correlation matrix. To perform an analysis at the seed level, information about the main activation sites of a certain RSN must be provided. After identifying the seeds belonging to a RSN, then co-activations seed maps can be built by overlapping the correlation maps of each seed (Fox et al., 2005). The brain regions that correlate with all the seeds from a RSN are identified as part of that RSN. Performing the correlation matrix between signals (estimated either at the voxel or regional level) provides information regarding all pair correlations in the brain. Each line in the matrix corresponds to a seed, and the entries in that line correspond to the correlation coefficient between the activation level in that seed and all the remaining regions (columns). One advantage of computing correlation matrices is that they can be studied using graph theory, not only to evaluate the topological properties of functional networks, but also for the detection of functional modules and hubs. Investigating correlation matrices at the voxel level can be computationally costly so it is common to average the signals from voxels falling in the same cortical region (defined according to a parcellation scheme), and then analyze the correlation matrix at a much lower resolution (as inHoney et al., 2009). Moreover, wavelet analysis can be applied to the temporal signals to compute frequency-dependent correlation matrices. For example, this approach allows searching for the frequency range over which maximal differences are observed between healthy controls and subjects (Bassett et al., 2012; Lynall et al., 2010; Supekar et al., 2008). Despite its usefulness for detecting linear statistical dependencies, the correlation analysis has certain limitations. The most important relies on the fact that RSNs are not spatially independent and can overlap. In other words, the same cortical region can belong to more than one RSN at a time and therefore activates whenever one or another RSN is engaged. In this way, the activation pattern of that region turns out to be a sum of the activation patterns of each RSN it belongs to, which cannot be captured using correlation-based approaches (Smith et al., 2012). Recent studies propose the use of ICA to extract RSN spatial maps from coordinated BOLD fluctuations (Beckmann, 2012; Beckmann et al., 2005; Damoiseaux et al., 2006; De Luca et al., 2006; Mantini et al., 2007). ICA is a computational technique for identifying hidden statistically independent sources from multivariate data. Importantly, ICA can be spatial (optimizing for spatial independence between components) or temporal (optimizing for temporal independence between components). In other words, spatial ICA allows mapping the areas in the brain that consistently activate together, while temporal ICA maps the regions in the brain that mostly contribute to a given temporal signal. To date, nearly all applications of ICA to fMRI (including resting-state fMRI) have used spatial ICA, because ICA requires a large number of samples to function well, and in fMRI there are orders of magnitude more voxels than time points. However, for finding temporally independent, potentially spatially overlapping, functional networks, temporal ICA provides a better approach (Boubela et al., 2013; Smith et al., 2012). Applying ICA to resting-state signals relies on the assumption that brain activity during rest results from the additive combination of independent spatial signals. ICA is a particular case of blind source separation and the signal in each pointxn (t),n=1,...,Ncan be written as a composition of the different independent componentsICk (t),k=1,...,Kwith coefficientsan,k in the following way:x n ( t ) = a n , 1 I C ( t ) 1 + a n , 2 I C 2 ( t ) + ... + a n , k I C ( t ) k An important note to consider in ICA is that the number of independent componentsKis, in theory, equal to the number of sourcesNbecause ICA cannot sort (nor scale) the source signals in correct order. However, reducing the number of sources (or dimensions) is possible by previously performing principal component analysis (PCA). PCA performs a linear mapping of the data to a lower dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, it is obtained by computing the eigenvalue decomposition of the covariance matrix of the signals. Then, the eigenvalues are sorted according to the proportion of variance they account for. Dimension reduction is possible by selecting the eigenvalues (i.e. principal components) that represent most of the variance. Beckmann et al. (2005)used probabilistic-ICA (PICA) to detect RSNs that exhibit high spatial consistency across subjects and closely resemble discrete cortical functional networks (such as visual cortical areas or sensory-motor cortex). Later,Damoiseaux et al. (2006)used the same technique and identified 10 robust RSNs across healthy subjects with potential functional relevance, consisting of regions known to be involved in motor function, visual processing, executive functioning, auditory processing, memory, and the DMN (seeFig. 2). Moreover, they report these networks exhibit significant baseline dynamics, with percentage BOLD signal changes up to 3% (comparable with the signal changes found in task-related experiments). The extraction of RSNs from whole-brain resting-state BOLD signals using ICA is particularly important because it does not require any prior information regarding the expected location of RSNs (seeFig. 10).Brookes et al. (2011)applied ICA to the amplitude envelopes of band-passed source-projected MEG signals and detected 8 IC spatial maps that closely matched the RSNs found with fMRI. These ICs were estimated from the envelopes of alpha- and beta-frequency oscillations (seeFig. 4). Other methods for characterizing resting-state fMRI networks include partial correlation (Fransson and Marrelec, 2008), coherence and partial coherence (Salvador et al., 2005b), and phase relationships (Sun et al., 2005). Both correlation and IC analysis rely on the linear relationship between regions. Other nonlinear functional connectivity measures from information theory, such as mutual information, have been applied to resting-state data (Hartman et al., 2011; Hlinka et al., 2011). Hlinka and colleagues defined functional connectivity matrices using both linear (Pearson correlation) and non-linear functional connectivity measures and evaluated the resulting networks using graph theory (Hlinka et al., 2011). Their results show that, at least from a graph-theoretical perspective, the nonlinearities in resting-state activity are practically negligible when compared to the inter-subject variability of the graph measures. On the group-average level, the nonlinearity effects are unnoticeable as they vary individually. In addition Lynall and colleagues analyzed resting-state data using both mutual information and correlation measures at different wavelet scales, and found that correlation measures allowed a better distinction between healthy controls and schizophrenia patients (Lynall et al., 2010). The linear dependency of resting-state BOLD time-series is probably linked to the low temporal resolution typically used in BOLD-fMRI studies (i.e. in the order of a couple of seconds), which does not allow the detection of fast non-linear interactions. Recent advances in ultra-fast spatial encoding and inverse reconstruction have increased the temporal resolution of whole-brain fMRI to time scales on the order of 100ms and faster. This unprecedented temporal resolution has been shown to significantly increase BOLD sensitivity resulting in improved detection of single trial task activation and resting state networks(Posse et al., 2012, 2013). 3.4 Characterizing brain networks using graph theory The first approach to capture the organizational properties of brain networks is to model them as graphs whose nodes represent dynamical units and whose links stand for the connections (structural or functional) between them. A large number of measures have been defined in the field of graph theory to characterize the topology of graphs (Boccaletti et al., 2006). Moreover, the availability of free Matlab libraries for graph analysis of brain networks, such as the Brain Connectivity Toolbox (www.brain-connectivity-toolbox.net,Rubinov and Sporns, 2010) and the MatlabBGL Toolbox (www.cs.purdue.edu/homes/dgleichby David Gleich) has allowed neuroscientists to efficiently implement the graph algorithms to explore the organization of brain networks. A brain connectome, be it anatomical or functional, withNcortical regions can be represented in the form of a matrixC, withNcolumns andNrows, where each entry in the matrixC(n,p) orCnp encodes the connection strength between two brain areasnandp(withn,p=1,...,N). Connections can be binary, simply denoting whether a link is present or absent. However, in brain networks, the connection is usually weighted: in structural networks, it is generally scaled by the number, density, or coherence of white matter fibre tracts; in functional networks it may corresponds to magnitudes of correlational or causal interactions between the two brain areas. Studying weighted graphs, however, imposes an increased degree of complexity and most graph theoretical studies of resting-state functional networks have used binary graphs instead. To transform the connectivity matrixCinto a binary adjacency matrixA, one needs to define a threshold,th, whereAnp =1 ifCnp >th, orAnp =0 otherwise. The definition of this threshold has direct impacts on the density of connections (or sparsity) of the network, which may have non-negligible effects on the graph properties of the network (Bassett et al., 2012; van Wijk et al., 2010). Therefore, to compare graph measures between networks of different subjects, it is usual to define equi-sparse graphs (instead of equi-threshold graphs), to ensure the same number of edges in all graphs (Bassett et al., 2012; Lynall et al., 2010) (seeFig. 11). Another crucial step on the definition of brain graphs is the size of the network, which is taken as the number of nodes. Since the brain can be parcellated at different scales (such as neurons, voxels or cortical regions) different graph properties may arise, and therefore the properties of brain networks must be taken in the light of the parcellation and the thresholding technique employed. Even so, two review studies on human brain graphs (Bassett and Bullmore, 2009; He and Evans, 2010) reported that some complex network properties are consistent over a range of spatial and time scales, and across modalities of neuroimaging data. Conserved principles include small worldness, high efficiency/low wiring cost, modularity and hubs. These properties were consistently found in brain networks obtained with both structural MRI (Bassett et al., 2008; Chen et al., 2008; He et al., 2007), diffusion MRI (Hagmann et al., 2008; Iturria-Medina et al., 2008), fMRI (Ferrarini et al., 2009; van den Heuvel et al., 2008; Wang et al., 2009a), EEG (Micheloyannis et al., 2009; Rubinov et al., 2009), and MEG (Stam et al., 2009; Valencia et al., 2008). The small-world architecture is particularly important in functional networks because it supports both segregated modular specialization and distributed functional integration (Sporns and Honey, 2006). In addition, it maximizes the efficiency of information transfer at a relatively low wiring cost. Notably, it seems to be a common self-organizing principle of complex natural systems, such as biological, chemical and even social networks (i.e. the famous "six degrees of separation" postulated between living people, derived from the original experiments by Milgram,Milgram, 1967). The importance of a small-world topology for an optimal cognitive performance is corroborated by reports of disrupted small-world properties in resting-state functional networks of people with schizophrenia (Bassett et al., 2012; Liu et al., 2008; Lynall et al., 2010), Alzheimer's disease (Supekar et al., 2008) and attention-deficit/hyperactivity disorder (Wang et al., 2009b). Another important graph measure is the modularity, which identifies modules of linked nodes that work together to achieve distinctive functions (Newman, 2006; Leicht and Newman, 2008). Connections are usually denser within modules than between them. Detecting and characterizing modules of the brain can allow us to identify groups of anatomically and/or functionally associated components that may subserve specific behavioural functions. For example, the modularity of resting-state fMRI functional networks has been found to change with normal ageing (Meunier et al., 2009). Note however that these changes may not be directly related to neuronal activity but to heart function and vessel compliance changes, naturally occurring in normal ageing. In a model of resting-state dynamics,Deco et al. (2009)used a modularity algorithm to divide one hemisphere of the macaque anatomical network into 2 communities (see more details later). Exploring the dynamics of the two modules independently, they found that each module exhibited a specific temporal pattern of synchronization, and notably, the synchronization level between the two modules was anti-correlated, which could be explained by stochastic resonance in a system with multistability. These results reinforce the idea that the topological organization of structural brain networks constrain and mould brain dynamics at multiple levels. 3.5 Models of brain network development and organization In the rapidly growing field of brain network science, some models have been proposed to explain the evolutionary organization principles of brain networks and their implications in brain function. For example, studies of anatomical and functional brain networks have shown that the connection probability between two brain regions decreases with the physical distance between them (Chklovskii and Koulakov, 2004; Salvador et al., 2005a). Based on this observation, generative models can be built by defining the probability of a functional connection (edge) between two cortical regions (nodes) separated by some Euclidean distance in anatomical space (Kaiser and Hilgetag, 2004a,b; Vertes et al., 2012). Such models may be optimized to reflect graph-theoretical aspects of brain networks, such as high efficiency or high clustering coefficient, leading to generative models of networks with small-world organization, like the brain. The economic network model fromBullmore and Sporns (2012)addresses how the brain optimizes an economic trade-off between the cost (i.e. minimizing connection density) and the efficiency of network function (i.e. minimizing the characteristic path length). Another theoretical model (Vertes et al., 2012) proposes that the topology of human brain functional networks emerges from two competing factors: lower probability of long-range connections and higher connection probability between regions sharing similar input. Although such models allow investigating certain organizational features of resting-state functional connectivity, they do not actually model the neural activity (i.e. dynamics) of interacting brain areas nor investigate the neurophysiological and/or biophysical mechanisms coupling brain areas together. As such, the network dynamics underlying brain activity during rest may not be fully explored using these models. 4 Large-scale models of resting-state dynamics The dynamics emerging spontaneously from the interplay between brain areas when these are embedded in the neuroanatomical network, has been attracting a growing body of research in computational neuroscience (Deco et al., 2011, 2013b; Jirsa et al., 2010; Nakagawa et al., 2013; Ritter et al., 2013). By means of whole-brain network models, constrained by realistic structural connectivity, one can explore how large-scale interactions give rise to the spontaneous emergence of resting-state fluctuations. Importantly, these models can be used to test predictions and evaluate the impact of lesions in the brain. The recently launched Virtual Brain (www.thevirtualbrain.org) is a neuroinformatics platform with a user-friendly brain simulator which allows users to perform customized simulations, analyze the results and compare them with macroscopic neuroimaging signals (Ritter et al., 2013). To investigate the spontaneous dynamics of interacting brain areas at the whole-brain scale, it is useful to go beyond the microscopic activity of individual neurons and consider instead the mesoscopic behaviour of large ensembles of neurons, or neuronal populations (Deco et al., 2008). Although simulations of detailed models at the cellular (and even sub-cellular) level are becoming computationally feasible (Izhikevich and Edelman, 2008; Markram, 2006), reduced neural-mass models, despite their low spatial resolution, allow a comprehensive study of the large-scale interactive dynamics with relatively low parametric complexity. This approach is motivated by neuroimaging observations showing that neurons within a densely connected neural ensemble tend to share the same physiological properties, exhibit dense reciprocal interconnectivity and show strong dynamical correlations. Neural-mass models can be extended to neural-field models, where the expected state of a neuronal population becomes a function of both time and position on the brain's spatially continuous cortical sheet. To date, however, all the models of whole-brain resting-state dynamics have represented brain areas as isolated points in space (or point-masses). Following differentreduction lines, a number of studies have contributed for the understanding of resting-state dynamics using neural-mass models coupled according to the brain's anatomical architecture (Cabral et al., 2011; Deco et al., 2009; Deco and Jirsa, 2012; Ghosh et al., 2008c; Honey et al., 2007, 2009). All these models rely on the assumption that resting-state activity arises solely from synaptic interactions between brain areas and disregard the role of physiological signals (such as blood flow, vessel structure and cerebrospinal fluid) which vary strongly across the human brain and even display very slow (<0.1Hz) fluctuations (Chang et al., 2013; Kim and Ogawa, 2012; Moser et al., 1999; Strik et al., 2002). In the approach followed by these models, the BOLD signal is obtained from the simulated neural activity using a simple model (i.e. the Windkessel-Balloon model applied uniformly to all brain areas (see Section4.7). In the following, we review existing models of resting-state activity. 4.1 Conductance-based biophysical model The first work to investigate the cooperative behaviour of neural systems coupled through a realistic anatomical wiring scheme was achieved byHoney and colleagues (2007). They used a biophysical neural-mass model introduced byBreakspear et al. (2003)together with the anatomical connectivity of the macaque cortex (Kotter, 2004). Later, the same model was extended to incorporate human neuroanatomical connectivity and results were compared with resting-state fMRI functional connectivity from healthy humans (Honey et al., 2009). Finally, the same group used this model to study the dynamical impact of lesions in the brain (Alstott et al., 2009; Honey and Sporns, 2008). The neural-mass dynamics in these works was derived from a conductance-based model of neuronal dynamics (Morris and Lecar, 1981) extended for neural population activity (Larter et al., 1999). The coupling between neural masses (or cortical regions) was introduced via weak long-range excitatory-to-excitatory connections, mimicking glutamate-induced synaptic currents (Breakspear, 2004; Breakspear et al., 2003). The main dynamical variable in that model is the mean membrane potential of pyramidal cellsV, which is governed by the conductance of sodium (gNa ), potassium (gk ) and calcium (gCa ) ions through voltage-gated channels, plus the passive conductance of 'leaky' ions (gL ). The total current flow across pyramidal cell membranes is given by:C d V d t = - g C a m C a ( V - V C a ) - g N a m N a ( V - V C N a ) - g K W ( V - V K ) - g L ( V - V L ) wheregion is the maximum conductance of each population of ions,mion is the fraction of open ion channels (Wfor potassium ion channels), andVion is the Nernst potential for that ion species. All equations and parameters are non-dimensional and normalized to neural capacitanceC=1. Each voltage-gated channel opens when the membrane potential overcomes a given threshold,Tion . For a large population of ion channels,Tion assumes a Gaussian distribution (with variancedeltaion ) and hence, the fraction of open ion channels is given by the following (sigmoid-shaped) function:m i o n = 0.5 1 + tanh V - T i o n delta i o n . The fraction of open potassium channelsWis defined differently because these channels 'relax' from one state to another at an exponential rate. Therefore,Wis governed byd W d t = phi ( m K - W ) tau wherephi is a temperature scaling factor and istauthe 'relaxation' time constant. To introduce synaptic interactions between neurons within the same neural ensemble, the average firing-rates of excitatory( Q V ) and inhibitory neurons( Q Z ) is calculated and introduced as a feedback term subsequent to cell firing to represent neurotransmitter release. At the cell soma, the membrane potential triggers an action potential if it exceeds a threshold. Averaging this over the ensemble of neurons and assuming once again a Gaussian distribution, the cell firing rates can be obtained by the following equations:Q V = 0.5 x Q V max 1 + tanh V - V T delta V , Q Z = 0.5 x Q Z max 1 + tanh Z - Z T delta Z , whereQmaxis the maximum rate of firing of the excitatory or inhibitory neurons. The firing of each population feeds back onto the ensemble and raises or lowers the membrane potential accordingly. Excitatory-to-inhibitory and inhibitory-to-excitatory connections are modelled as additional inputs to the flow of ions across the membrane channel, weighted by functional synaptic factors,aei andaie respectively. In addition, excitatory-to-excitatory connections are modelled with greater physiological detail: the mean firing rateQV is assumed to lead to a proportional release of glutamate neurotransmitter across the synapse, which diffuses onto two classes of ligand-gated ion channels. On one side, AMPA receptors open additional sodium channels, increasing the net conductance of sodium flow. On the other, NMDA receptors open an additional population of voltage-gated calcium channels, increasing the maximum conductance of voltage-gated calcium channels. Incorporating these specifications, the membrane potential of excitatory (V) and inhibitory (Z) is given by:d V d t = - ( g C a + r N M D A a e e Q V ) m C a ( V - V C a ) - ( g N a m N a + a e e Q V ) ( V - V N a ) - g K W ( V - V K ) - g L ( V - V L ) + a i e Z Q Z + a n e I delta , d Z d t = b ( a i i I delta + a n i V Q V ) , whereIdelta corresponds to 'nonspecific' subcortical excitation with amplitudeImodulated by a random noise component of amplitudedeltaadded to both populations with weightsane andani .aee scales the local excitatory-to-excitatory synaptic strength andrNMDA denotes the number of NMDA receptors relative to that of AMPA receptors. These equations depict the behaviour of one population of densely interconnected excitatory and inhibitory neurons. To model interaction ofNcoupled neural-masses it is necessary to introduce long-range excitatory projections between pyramidal cells. These long-range projections are modelled to target the same populations of NMDA and AMPA receptors targeted by the short-range excitatory projections. Representing each node with an indexn=1,...,N, the following equation describes the mean membrane potential of pyramidal cells at positionxn :d V ( x n ) d t = - ( g C a + ( 1 - c ) r N M D A a e e Q V ( x n ) + c r N M D A a e e Q V ( x ) ) m C a ( V ( x n ) - V C a ) - ( g N a m N a + ( 1 - c ) a e e Q V ( x n ) + c a e e Q V ( x ) ) ( V ( x n - V N a ) - g K W ( V ( x n ) - V K ) - g L ( V ( x n ) - V L ) + a i e Z Q Z ( x n ) + a n e I delta , where the notation< > represents spatial averaging over neural-masses. Parameters were set to values that replicate realistic conductances. Using a regular connectivity these values had previously been reported to show complex, spontaneous activity, including intermittency, phase synchrony, and marginal stability (Breakspear et al., 2003; Breakspear, 2004). Spontaneous brain activity was simulated using realistic anatomical connectivity, first from the macaque brain (Honey et al., 2007) and later from the human brain (Honey et al., 2009). The inter-node coupling was set to a value at which synchronous dynamics is weakly stable, allowing spontaneous switching between synchronous epochs and desynchronous bursts. Since neither noise nor delays are introduced in the model, activity in the system arises purely from nonlinear instabilities due to the complex structural connectivity and the chaotic dynamics at the neural-mass level. Under this parameterization the neural interactions in the model occur at multiple time-scales and reflect spontaneously arising 'self-organizing' patterns. Using the Balloon-Windekessel model (Friston et al., 2003), the simulated activity of each pool was transformed into a BOLD signal. Results show a spatial organization of BOLD signal fluctuations, which favourably compares to empirical resting-state functional connectivity from 5 healthy subjects (Honey et al., 2009). These results were the first to indicate that the spatio-temporal characteristics of resting-state activity are constrained by the large-scale anatomical structure of the human cerebral cortex. 4.2 The FitzHugh-Nagumo model The neuroanatomical architecture of the brain shapes not only the connectivity between regions, but also the distance over which these connections occur, defining in this way a space-time structure of couplings and delays, which is essentially constant over relatively short time scales. Ghosh and colleagues explored the time-delayed interaction between neural-masses in this space-time structure, with delays proportional to the distance between regions (Ghosh et al., 2008a,c). The ongoing dynamics was simulated using FitzHugh-Nagumo units (Fitzhugh, 1961; Nagumo et al., 1962) coupled according to anatomical connectivity of one hemisphere of the macaque brain (Kotter, 2004). The implementation of the model consisted in defining two state variables,u n andv n , representing the membrane potential and the recovery potential correspondingly, for each noden. A parametercscales all connection strengths without altering the connection matrixC n p nor affecting the associated time delaysDelta t n p . The following differential equations describe the dynamics of the state variables, with corresponding additive noisen u andn v :u ? n ( t ) = g ( u n , v n ) - c ? n = 1 N C n p u p ( t - Delta t n p ) + n u ( t ) v ? n ( t ) = h ( u p , v n ) - n v ( t ) The functionsgandhare based on FitzHugh-Nagumo systems (Fitzhugh, 1961; Nagumo et al., 1962) and are defined as follows:g ( u n , v n ) = tau v n + gamma u n - u n 3 3 , h ( u n , v n ) = - 1 tau [ u n - alpha + beta v n ] In absence of connectivity, the network nodes display damped oscillatory dynamics. Nodes were then coupled with realistic brain connectivity and delays and the stability of the resulting dynamics was investigated as a function of finite signal transmission speeds and increasing couplingstrengths. For a restricted interval of velocities (5-20m/s) and for a sufficiently strong coupling, the system exhibits increased instability. At the border of instability, and in the presence of noise, neural-masses display the emergence of oscillatory dynamics (at 10Hz) with fluctuating amplitude (seeFig. 12). The structural connectivity is found to shape the dynamic repertoire of the entire system, giving rise to slow fluctuations in the power of the 10Hz oscillations. To test for the emergence of ultra-slow oscillations, the BOLD signal was computed (see Section4.7), giving rise to rest-like fluctuations that were correlated across distant areas. Furthermore, simulated BOLD signals were found to exhibit correlations within the DMN as transmission velocities ranged between 5 and 10m/s, which fall in the interval of expected realistic speeds in myelinated axons. In addition to the results presented in their study, the authors have tested multiple oscillator types which are commonly used in neural-mass modelling including Hopf oscillators, Wilson-Cowan systems, FitzHugh-Nagumo systems, and finally mixed populations of coupled FitzHugh-Nagumo neurons (Assisi et al., 2005), all providing similar results, reinforcing the key role of the space-time structure of the neuroanatomical network in shaping the dynamics of the brain at rest. 4.3 The Wilson-Cowan model Another important modelling study of large-scale resting-state dynamics was developed byDeco and colleagues (2009)using the same space-time structure as (Ghosh et al., 2008a), i.e. the macaque's anatomical connectivity (CoCoMac;Kotter, 2004) with time delays derived from a human template. At the node level, the dynamics of a neural population was modelled using Wilson-Cowan units (Wilson and Cowan, 1972, 1973). Wilson and Cowan analyzed the collective properties of large ensembles of excitatory and inhibitory neurons using methods from statistical mechanics, based on themean-fieldapproach. They proposed that typical dynamics in a cortical region could be obtained by considering a population of excitatory neurons coupled with a population of inhibitory neurons (Fig. 13A). It consists of a set of differential equations that describe the time evolution of the mean level of activity of a neural population, using a nonlinear sigmoid function to represent the interactions between the populations. The activity of a pool of excitatory (pyramidal) neurons without external input,E(t), is given by the following equation:tau ? E ( t ) ? t = - E ( t ) + phi ( E ( t ) ) , wheretauis the membrane time constant, -E(t) means that the activity decays in time if no stimulation is received, and the last term takes into account the recurrent excitatory stimulation from all the neurons in the same pool. The response function transforming the current into discharge rates is given by:phi ( x ) = c 1 - exp - a ( x - b ) . Considering now the two populations of the Wilson-Cowan module are coupled together (Fig. 13A), one with only excitatory neurons,E ( t ) , and another with only inhibitory neurons,I ( t ) , the dynamics of such module is given by the following 2 differential equations:tau ? E ( t ) ? t = - E ( t ) + phi ( I b + w + x ( t ) - I ( t ) ) + delta ( t ) , tau ' ? I ( t ) ? t = - I ( t ) + phi ( w I E ( t ) ) + delta ( t ) , whereI b is a diffuse spontaneous background input, anddelta is additive independent Gaussian noise with mean value zero. The noise level is given by the variancedelta 2 . To model the network dynamics at the macroscopic level,N = 38 Wilson-Cowan modules, each representing a cortical region, were coupled according to the macaque's neuroanatomical connectivity, using the connectivity matrixC n p and the delaysD n p . The global dynamics is given bytau ? E n ( t ) ? t = - E n ( t ) + phi I b + ? p = 1 N alpha C p n E p ( t - D p n ) - I n ( t ) + delta n ( t ) , andtau ? I n ( t ) ? t = - I n ( t ) + phi ( w I E n ( t ) ) + delta n ( t ) . alpha regulates the global coupling strength and recurrent couplings were set toC n n = w + / alpha . Taking into account the bifurcation diagram of a single Wilson-Cowan module as a function of the parametersI b andw + , these 2 parameters were selected such that the node dynamics is at the border of a Hopf bifurcation, but still in a non-oscillatory low-activity state. The idea is that, when modules are coupled (alpha > 0 ), they go spontaneously to an oscillatory state in the gamma-frequency band (40Hz). Notably, for a range of couplings, time delays, and noise, each module, when coupled, behaves similarly to single isolated and noise-free Wilson-Cowan in the oscillatory regime. As described earlier, using a modularity algorithm (Newman, 2006), the structural network of the macaque brain was divided into 2 modules, with more probability of links within the modules than between them. For a certain range of parameters (couplings, delays and noise), the two modules tend to synchronize alternatively, while the network is never globally synchronized. This dynamical regime, where 2 modules compete to synchronize, is called a Chimaera state in theoretical physics. This leads to slow anti-correlated fluctuations in the synchrony degree of the two different modules. For an optimal level of noise, the synchronization level of the two clusters was found to fluctuate slowly (0.1Hz) in the same time-scale of the BOLD signal fluctuations observed experimentally during rest. This work (Deco et al., 2009) suggests a theoretical scenario for the origin of anti-correlated BOLD signal fluctuations observed in the brain at rest (Fox et al., 2005). However, posterior studies have shown that the anti-correlations in BOLD signal reported byFox et al. (2005)may be introduced via certain pre-processing approaches, in particular the regression against the global signal, far from the biophysical synchronization mechanisms proposed in this work (Murphy et al., 2009; Weissenbacher et al., 2009). 4.4 The Kuramoto model Following the results fromDeco and colleagues (2009), where neural populations were intrinsically oscillating in the gamma-frequency band, an increased degree of reduction was applied to the model to deeper investigate the role of local oscillations in resting-state functional connectivity (Cabral et al., 2011). The interaction of oscillatory cortical regions in the brain's large-scale anatomical network was simulated using a well-known model describing coupled oscillators' systems, called the Kuramoto model (Acebron et al., 2005; Kuramoto, 1984). The Kuramoto model has been used to simulate synchronization behaviour in a wide variety of fields, including biological systems and, more recently, neural dynamics (Breakspear et al., 2010). The reduction of a neural-mass dynamics to a phase oscillator is made in an abstract way, supported by a number of experimental and theoretical studies showing that neural activity at the population level usually exhibits oscillations with a moderate level of synchrony, due to a balance in the firing rates of excitatory and inhibitory neurons, mainly in the gamma frequency range (30-80Hz) (Bartos et al., 2007; Borgers and Kopell, 2003; Brunel, 2000; Brunel and Wang, 2003). According to the Kuramoto model, the behaviour of coupled oscillators in nature can be modelled as a sine function of the phase difference between the two oscillators. Without delays, coupled oscillators tend naturally towards synchronization, as long as the coupling is sufficiently strong to engage phase interactions. However, in the presence of delays, the dynamics becomes more complex (Yeung and Strogatz, 1999) and the particular case of heterogeneous delays, like in the brain, has only recently begun to be addressed theoretically (Lee et al., 2009). Denoting bytheta n ( t ) the phase of nodenat timet, it then obeys the following dynamical equation:d theta n d t = omega n + k ? p = 1 N C n p sin ( theta p ( t - tau n p ) - theta n ( t ) ) , n = 1 , ... , N . whereC n p is the structural coupling strength from nodepto noden, andkis the global coupling strength which scales all connection strengths. The delaytau n p between nodepand nodenis calculated usingtau n p = L n p / v , whereLnp is the distance between nodes andvis the conduction velocity.f n = omega n / 2 pi is the intrinsic frequency of noden. At the network level, the synchrony of a group of nodesNcan be evaluated by the order parametersR(t) andPhi(t), jointly defined byR ( t ) e i Phi ( t ) = 1 N ? n = 1 N e i theta n ( t ) , whereR(t) measures phase uniformity and varies between 0 for a fully desynchronized state and 1 for a fully synchronized state, andPhi(t) represents the phase of the global ensemble. For a biologically plausible range of time delays (between 8 and 15ms) and sufficiently strong coupling, the system is in a regime where subsets of nodes tend to synchronize although the network is not globally synchronized. This unstable synchronization of subsets of brain areas (or functional networks), leads to slow fluctuations in the synchrony degreeR(t). The intensity of these fluctuations is given by the standard deviation ofR,sigma R , which indicates the degree of metastability of the system (Shanahan, 2010). In the model, this metastable synchronization originates slow fluctuations in neural activity that are captured by the BOLD signal (see Section4.7). Notably, the simulated BOLD signal correlates between brain areas and the correlation patterns are in good agreement with the empirically measured BOLD correlation structure (seeFig. 14). This modelling work shows that resting-state activity can originate from oscillatory network interactions, where different subnetworks, defined by the space-time structure of the anatomical network, tend to synchronize temporarily while the global system is never fully synchronized. In this theoretical scenario, resting-state functional networks are subnetworks thattemporarily synchronize when the global brain network operates in the border between incoherence and synchrony. These functionally relevant subnetworks are intrinsically defined by the neuroanatomical network. 4.5 Node model in asynchronous state It is now widely accepted that the dynamics of the brain at rest is strongly shaped by the underlying anatomical network. However, the effects of structural disconnection in resting-state fMRI functional connectivity remain unclear.Cabral et al. (2012a)approached this issue using, at the node level, a simplified model consisting in a linearized version of the Fokker-Planck equation. Differently from the Kuramoto model described previously, this model builds on the assumption that local neural networks are in a stable asynchronous state where no oscillations develop. In this case, the node model describes only the firing-rate deviations around the steady state. At rest, these deviations are induced by internally generated noisy fluctuations (finite size networks have been shown to intrinsically induce noise in the dynamics,Mattia and Del Giudice, 2002). In the presence of noise, the model produces ongoing fluctuations by the excitatory reverberation of local activity on the large-scale structural network. Consequently, the structural connectivity makes the rate fluctuations not independent, explaining in principle the finding of the large-scale neuroanatomical structure in brain fluctuations. According to the Fokker-Planck equation (Risken, 1989), the activity of a large population of spiking neurons can be described by the probability distribution of the neuron's internal variables (i.e. the membrane potential in the simplest case). An associated equation gives the neural population firing-rate. Under the hypothesis that the neural populations are in a stable asynchronous state, the steady solution of the Fokker-Planck equation is stable and the probability distribution can be decomposed leading to a linear dynamical equation for each of the coefficients of the series' expansion plus an equation for the rate fluctuations. This model proposes that the resting-state BOLD signal fluctuations are generated by slow, noise-induced firing-rate deviations around the steady state and not by interactions of fast oscillatory signals (such as alpha or gamma-band oscillations as inCabral et al. (2011),Deco et al. (2009)andGhosh et al. (2008a)). As such, in this model, the fast oscillatory perturbations are neglected and only the slow frequencies captured by Balloon-Windkessel model (<0.4Hz) are considered. Therefore, the dynamics can be approximately described by the exponentially decaying perturbations responsible for the low-frequency part of neuronal activity. Consequently, for each brain arean, the firing rate deviationsr n ( t ) obey the following first order stochastic differential equation:tau 0 d r n d t = - r n ( t ) + k c 1 ? p = 1 N C n p r p ( t - tau n p ) + sigma eta n ( t ) , n = 1 , ... , N . wherekis the global excitatory coupling level between nodes (k > 0 ).C n p andtau n p are the structural coupling strength and the conduction delay from regionpto regionn, specified by the white matter connectivity.sigma is the noise level and the termseta n ( t ) are uncorrelated white Gaussian noises with zero mean and unit variance (< eta n ( t ) > = 0 and< eta n ( t ) eta p ( t ' ) > = delta n p delta ( t - t ' ) , wheredelta n p is the Kronecker symbol anddelta ( t ) denotes the Dirac delta function). As equations are linear,sigma only scales the level of the rate deviations. Since long-distance connections are excitatory, the reverberated activity over the network can destabilize the damped local dynamics, and therefore the asynchronous states. Giventau 0 , which is given by the internal state of local networks (tau0=20ms here), the dynamics depends only on one parameter:k. The parameterkwas manipulated to control the strength of excitatory coupling between brain areas, in order to study the global effects of structural connectivity. Simulations with the model generate slow BOLD signal fluctuations, whose correlations (i.e. the functional connectivity matrix) reveal the underlying anatomical connectivity (Fig. 15). For a certain coupling strength, simulations reproduced with good agreement healthy resting-state functional connectivity. 4.6 Attractor network of spiking neurons Recently, a new model of resting-state activity with an increased degree of realism was introduced (Deco and Jirsa, 2012). In this model, each area (or node) in the global brain network is modelled as a local spiking network consisting of mutually interconnected populations of excitatory pyramidal neurons and GABAergic inhibitory neurons. In other words, it is a network of networks, specified by a large set of coupled equations describing each neuron and synapse. The local networks consist of integrate-and-fire spiking neurons with excitatory (AMPA and NMDA) and inhibitory (GABA-A) synaptic receptor types (Brunel and Wang, 2001). Neurons in each local network were divided into two populations: one ofNE excitatory pyramidal and one ofNI inhibitory neurons. The neurons' spiking activity was described by the classical Integrate-and-Fire model, which specifies the membrane potentialV(t) as a function of the input currents coming from connected neurons and of external inputs. While the membrane potential of each neuron in the network is below a given thresholdVthr ,V(t) is given by the following equation:C m d V ( t ) d t = - g m ( V ( t ) - V L ) - I M - g A M P A , ext ( V ( t ) - V E ) ? j = 1 N ext S j A M P A , ext ( t ) - g A M P A , rec ( V ( t ) - V E ) ? j = 1 N E w j S j A M P A , rec ( t ) - g N M D A ( V ( t ) - V E ) / ( 1 + lambda e - beta V ( t ) ) ? j = 1 N E w j S j N M D A ( t ) - g G A B A ( V ( t ) - V I ) ? j = 1 N I w j S j G A B A ( t ) When the membrane potential crosses the thresholdVthr , the neuron generates a spike. The spike is transmitted to other neurons and the membrane potential is instantaneously reset toVreset and maintained there for a refractory timetauref , during which the neuron is unable to spike.gm is the membrane leak conductance,Cm is the capacity of the membrane andVL is the resting potential. The membrane time constant is defined bytau m = C m / g m . The synaptic input current is given by the last four terms of the equation, corresponding to glutamatergic AMPA (IAMPA,ext) external excitatory currents, AMPA (IAMPA,rec) and NMDA (INMDA ) recurrent excitatory currents, and GABAergic recurrent inhibitory currents (IGABA ) with the their respective synaptic conductancesgAMPA,ext ,gAMPA,rec,gNMDA andgGABA .VE andVI are the excitatory and inhibitory reversal potentials. The weight of recurrent self-excitationwj was given byw+=1.55 within excitatory populations, andw=1 within inhibitory populations. The connections between excitatory and inhibitory neurons were weightedw=1. The gating variablesS j I ( t ) represent the fraction of open ion channels on the membrane of neurons. ForI=AMPA or GABA receptor types, the gating variables are given by:? S j I ( t ) ? t = - s j I ( t ) tau I + ? k delta ( t - t j k ) . For NMDA synapses, the gating variables are specified as:? S j N M D A ( t ) ? t = - S j N M D A ( t ) tau N M D A , d e c a y + alpha x j N M D A ( t ) ( 1 - s j N M D A ( t ) ) ? x j N M D A ( t ) ? t = - x j N M D A ( t ) tau N M D A , r i s e + ? k delta ( t - t j k ) . The sums over the indexkrepresent all the spikes emitted by the presynaptic neuron j at timest j k .tau A M P A andtau G A B A are the decay times for AMPA and GABA synapses, andtau N M D A , r i s e andtau N M D A , d e c a y are the rise and decay times for the NMDA synapses. In addition, all neurons in the system receive an external background input representing the noisy fluctuations that are typically observed in vivo arising from finite-size effects of the spiking dynamics of the individual neurons. These fluctuations are represented by uncorrelated Poisson spike trains with a time-varying ratenu ext p ( t ) governed by:tau n ? nu ext p ( t ) ? t = - ( nu ext p ( t ) - nu 0 ) + sigma nu 2 tau n n p ( t ) , wheresigma nu is the standard deviation ofnu ext p ( t ) andn p ( t ) is normalized Gaussian white noise. Negative values ofnu ext p ( t ) that could arise due to the noise term are rectified to zero. All the values used in the simulations are reported inDeco and Jirsa (2012). The firing activity of the whole system was simulated and the BOLD-fMRI signal was estimated using the Balloon-Windkessel model (Friston et al., 2003). The simulated functional connectivity was computed as the correlation matrix of the simulated BOLD signals between all brain areas.Fig. 16(right) shows the Pearson correlation between both the empirical and the simulated functional connectivity matrices as a function of the global inter-area coupling weight. The large set of coupled equations describing each neuron and synapse in the global network specify a dynamical system. The stationary fixed points (or stable patterns of firing activity) of such system are called "attractors". To estimate the number of attractors of the system at low computational costs, a mean-field approximation was used (Brunel and Wang, 2001; Deco et al., 2013c). For each parameter set, the mean-field equations were iterated with 1000 different initial conditions chosen randomly for each node. The number of attractors was established as the number of final average firing rates that are different from all previously final average firing rates found for at least one pool. InFig. 16(left), the number of attractors was calculated for a range of global coupling strengthsW, providing the so-called "attractor landscape". For very small values ofW, only one attractor is stable. That attractor corresponds to the trivial spontaneous state of the system where all neurons in all brain areas display low firing activity. Also forvery large values ofWonly one stable attractor is found, but in this case it corresponds to the "epileptiform" case where all excitatory neurons are highly activated in all brain areas. Notably, for intermediate couplingW, multistability emerges and many attractors coexist, corresponding to distinct foci of high firing activity in particular brain areas. The entropy of the system to characterize the expected variability of cortical activity due to noise-driven transitions between multistable attractors. When there is only one single attractor, the system invariably settles within it and the entropy is 0. If, however, the number of attractors is larger than 1, the entropy is given by:H = - ? i p ( i ) log ( p ( i ) ) , wherep(i) is the probability that the system settles in attractori. As shown inFig. 16(left), the entropy is increased for the range of coupling weightsWwhere multistability emerges. The model was found to optimally predict empirical data (Fig. 16, right) for a coupling weight where the brain network operates at the brink of the bifurcation that separates the stable low activity equilibrium state from the multistable state region where many attractors coexist (Fig. 16, left). The results from this work demonstrate that resting-state networks in fMRI can result from structured noise fluctuations around the trivial low firing equilibrium state induced at the edge of a bifurcation by the presence of latent "ghost" multistable attractors corresponding to distinct foci of high firing activity in particular brain areas. Importantly, this work proposes a new scenario where the multistable attractor landscape, which is inherently present in the neuroanatomical connectivity, defines a functionally meaningful dynamic repertoire of the brain network. These functionally relevant attractor networks can be rapidly activated during a task through the attractor's stabilization since, during rest, the brain operates at the edge of the bifurcation. The dynamic mean field model (Deco et al., 2013c) approximates the ensemble dynamics of the detailed spiking large-scale network reducing its complexity. The model was further simplified into a set of motion equations for statistical moments, providing a direct analytical link between anatomical structure and functional connectivity. With this reduction, the authors show that, in this model, the resting-state activity emerges as structured linear fluctuations around a stable low firing activity state close to destabilization. 4.7 Transforming neuronal activity into BOLD signal To compare the simulated neural activity with data from resting-state fMRI functional connectivity, it is necessary to estimate the BOLD signal changes associated with the simulated neural activity. The neurovascular transduction is defined by the haemodynamic response function which describes the way blood flow adjusts to deliver oxygen to active neuronal tissues. The Balloon-Windkessel haemodynamic model (Friston et al., 2003) is widely used to model the neurovascular coupling and has been used in all existing models of whole-brain resting-state activity (Cabral et al., 2012a,b; Deco et al., 2009; Deco and Jirsa, 2012; Ghosh et al., 2008a; Honey et al., 2009). Although the same biophysical parameters have been used for all areas, the shape and time to peak of this function may vary substantially between brain areas and even across subjects (Aguirre et al., 1998; Cunnington et al., 2002; D'Esposito et al., 1999; Handwerker et al., 2004). These variations may be further increased with age, medication and pathologies. As such, assuming consistency across subjects or brain regions (and even to model disease) is a considerable simplification. In the Balloon-Windkessel model, the BOLD signal is taken to be a static nonlinear function taking into account the normalized voxel content of deoxyhemoglobin, the normalized venous volume, the net oxygen extraction fraction by the capillary bed during rest and the resting blood volume fraction. The neural activity at noden,r n ( t ) , causes an increase in a vaso-dilatory signals n that is subject to auto-regulatory feedback. Inflowf n responds in proportion to this signal with concomitant changes in blood volumev n and deoxyhemoglobin contentg n . The equations relating these biophysical variables with the BOLD signalgamma n are:? s n ( t ) ? t = r n - k n s n - gamma n ( f n - 1 ) ? f n ( t ) ? t = s n tau n ? v n ( t ) ? t = f n - v n 1 / alpha tau n ? q n ( t ) ? t = f n ( 1 - ( 1 - rho n ) 1 / f n ) rho n - v n 1 / alpha q n v n whererho n is the resting oxygen extraction fraction. The BOLD signal is taken as a static nonlinear function of volume and deoxyhemoglobin that comprises a volume-weighted sum of extra- and intravascular signals:y n = V 0 ( 7 rho n ( 1 - q n ) + 2 1 - q n v n + ( 2 rho n - 0.2 ) ( 1 - v n ) ) whereV 0 = 0.02 is the resting blood volume fraction. The biophysical parameters were taken as inFriston et al. (2003). The Balloon-Windkessel model filters out the higher frequencies of neural activity and maximizes the amplitude of lower frequency oscillations (<0.5Hz with a peak at 0.2Hz using the biophysical parameters fromFriston et al. (2003), seeFig. 17B). In addition, the Balloon-Windkessel model also contains functions that transform neural activity into the BOLD signal in a non-linear way. As such, it is unclear whether the spontaneous BOLD signal fluctuations obtained in the resting-state models come directly from the low-frequency part of neural activity or result from non-linearities of the BOLD model. To test for this,Cabral et al. (2011), compared the output from the Balloon-Windkessel model with the low-pass filtered neural activity using a cut-off frequency of 0.35Hz, which showed the best agreement with the simulated BOLD signal (0.9 correlation with a lag of 1.6s) (seeFig. 17A-C). The time-lag between the low-pass filtered neural activity and the output of the Balloon-Windkessel model corresponds to the time to peak of the haemodynamic response function, which is defined by the time-constants of the model (Friston et al., 2003). These results indicate that the resting-state BOLD functional connectivity obtained in the models is mainly related to the very slow part of simulated neural activity and less related to non-linearities of the Balloon-Windkessel model. Please note that, although the network of vessels has not been considered in these models, the full arterio-venous passage takes about 3.5s, which is in the same range of the time to peak of the haemodynamic response function (in SPM the canonical time-to-peak is 5s). 4.8 Modelling the impact of structural lesions The complex wiring architecture of the neuroanatomical connectome has shown to play a fundamental role in the global characteristics of the brain at rest. As a consequence, damage to the structural connectome resulting either from stroke, traumatic brain injury or some neurological disease, may have impact on brain activity not only near the lesion site, but also at the macroscopic scale. As the prediction of lesion-induced changes is beyond reach on the experimental side, large-scale brain models are a unique tool to investigate the impact of such structural lesions (Alstott et al., 2009; Cabral et al., 2012a,b; Honey and Sporns, 2008). A structural lesion can be modelled with the removal of a node (brain area) or a link (structural connection) in the structural network. Using the biophysical model fromHoney et al. (2007),Honey and Sporns (2008)andAlstott et al. (2009)simulated the impact of brain lesions by removing nodes from the connectomes of the macaque (Honey and Sporns, 2008) and the human brain (Alstott et al., 2009). Findings indicate that lesions produce altered functional connectivity among distant regions of cortex, often affecting both cortical hemispheres (Fig. 18). These patterns are highly dependent on the location of the lesion in the network and its impact depends on the properties of the node (for example, its clustering coefficient). Motivated by studies reporting altered resting-state activity in people with schizophrenia, a disease for long associated with connectivity deficits (Friston and Frith, 1995), Cabral and colleagues investigated the outcome of a structural disconnection in the properties of resting-state functional networks. Using two different models at the node level (one where cortical areas are in a stable asynchronous state (Cabral et al., 2012a), and another where brain areas exhibit self-sustained oscillations (Cabral et al., 2012b)), both models focused essentially on the impact of a brain-wide decrease of the coupling strength in the properties of simulated resting-state functional networks. Importantly, the coupling strength in both models scales the excitatory-to-excitatory coupling between brain areas, which is ensured by brain mechanisms involved in long-range signal transmission in the brain. These mechanisms include both axonal connectivity (dependent on the number, density and coherence of axon fibres) and synaptic mechanisms (e.g. neurotransmission and plasticity). To test the performance of the model, the functional connectivity matrices were compared with experimental results from healthy controls and patients with schizophrenia (Lynall et al., 2010). To do the comparison, simulated functional networks were characterized using graph theory following the same methodology fromLynall et al. (2010). Simulated healthy functional networks were found to have graph properties in the range of the ones reported experimentally. When the structural connectivity was decreased, either globally or locally, the resulting simulated functional connectivity exhibited a network reorganization characterized by an increase in hierarchy, efficiency and robustness, a decrease in small-worldness and clustering and a narrower degree distribution, in the same way as recently reported for schizophreniapatients (Fig. 19). Theoretical results indicate that most disconnection-related neuropathologies (either at the global or local level) should induce the same qualitative changes in resting-state brain activity. 5 Discussion Resting-state activity has been investigated for almost 20 years now, but the exact origin of this activity remains unclear. To the extent that this activity is functionally relevant, the study of resting-state activity may provide a new light to understand brain function. In this review work, we brought together experimental findings from different neuroimaging techniques which provide pictures of resting-state activity from different perspectives (but also with different limitations). Overall, these studies point to the existence of a complex network dynamics emerging spontaneously during rest, with specific spatial, temporal and spectral characteristics. To explore this dynamics in detail, we started by describing how brain networks are defined, what are the differences between them, and which properties better characterize them. Subsequently, we have gathered together all the computational works that developed whole-brain models to investigate how the resting-state activity (with its particular spatio-temporal dynamics) emerges from the structural substrate of the neuroanatomical network. These models propose different theoretical scenarios to explain brain activity during rest, making different assumptions for the dynamics occurring at the time-scale of neural activity. As such, selecting the model that maximally fits the empirical BOLD functionally connectivity matrix is not sufficient to determine the best candidate. Instead, models need to be reviewed in the light of recent resting-state electrophysiological studies in order to select the scenario that best explains all the phenomena observed in resting-state activity at different temporal (and spectral) scales. 5.1 Experimental evidence of resting-state activity The most robust and prominent finding in the resting-state literature is the existence of slow and spatially organized BOLD signal fluctuations. Deeper analysis of these fluctuations have revealed specific spatio-temporal patterns, or RSNs, that activate and deactivate spontaneously on a slow time-scale (<0.1Hz). All of these patterns (except one) correspond to functional networks typically activated during goal-directed behaviour. The only exception is the DMN, which never activates during task, but consistently activates during rest in healthy adults. These findings seem to indicate a complex network dynamics emerging from intrinsic brain processes, but the mechanisms at the origin of this dynamics and its functional relevance remain unclear. The importance of these resting-state patterns for an optimal brain function is reinforced by the fact that they appear disrupted in normal ageing and disease. However, the contribution of (non-stationary) physiological noise and/or behavioural artefacts to these results cannot be totally ruled out. Several studies have aimed at exploring the electrophysiological counterpart of resting-state activity, in order to explore the mechanisms of resting-state functional connectivity. On one side, intracranial recordings reveal that local increases in the BOLD signal during rest are associated with local increases in the power of gamma-band oscillations. On the other side, simultaneous recordings of EEG and fMRI indicate that the BOLD signal is negatively correlated with the power of alpha- and beta-band idle rhythms. Given that the energy requirements of gamma-band oscillations (40-100Hz) are higher than alpha- and beta-band oscillations (8-13Hz and 13-30Hz) and since the BOLD signal reflects the metabolic demand of a given brain area, the two scenarios are not necessarily contradictory and could reflect different aspects of the same phenomenon. In addition, MEG studies have identified the same RSNs as in fMRI studies using the slow amplitude envelope fluctuations of alpha- and beta-band oscillations, which points to a relationship (not necessarily positive) between the BOLD signal and the power of these oscillations. However, it is important to mention that these studies are only preliminary and must be further validated. Indeed, RSNs have only recently begun to be explored using electrophysiology and the literature is scarce when compared to the vastness of resting-state fMRI studies. However, they are crucial for understanding the mechanisms behind resting-state activity and, therefore, must not be neglected. 5.2 Models of resting-state activity Coming up with a biophysical explanation that encompasses all the phenomena observed during the resting-state is not straightforward and the problem remains unsolved until today. Potential theoretical scenarios have been proposed and subsequently tested using bottom-up computational models. The general theoretical picture is that the resting-state activity emerges spontaneously from the interaction between brain areas in the neuroanatomical network. However, the spontaneous behaviour of brain areas has been modelled making different assumptions and following different reduction lines. Despite the differences, most models generate slow BOLD signal fluctuations with a correlation structure that fairly approximates the empirical functional connectivity map. 5.2.1 The anatomical connectivity One main component of resting-state models is the anatomical connectivity matrix, which expresses the large-scale wiring diagram of the brain. A common feature of the different models is that the functional connectivity is strongly shaped by the anatomical connectivity at the optimal working point. As such, anatomical matrices obtained using different techniques, algorithms or parcellation schemes lead to different results. For example, the macaque neuroanatomical matrix refers only to one hemisphere and links have only 3 scales of weighting (Deco et al., 2009; Ghosh et al., 2008a; Honey et al., 2007). On the other hand, the anatomical matrix fromHagmann et al. (2008)used in (Cabral et al., 2011; Deco and Jirsa, 2012; Honey et al., 2009) does not include subcortical routes that are known to play an important role in shaping the spontaneous activity of the brain (Freyer et al., 2011; Robinson et al., 2001). Therefore, a rigorous comparison between models would require testing all with the same anatomical connectivity, and comparing the results with the same functional connectivity. 5.2.2 The dynamical regime of brain areas The main difference between models is the type of dynamics assumed at the node level, i.e. the intrinsic behaviour of a brain area when it is in the spontaneous state. In general, three major stationary regimes have been proposed for an isolated brain area (seen as a neural ensemble): a fixed-point attractor, a limit-cycle attractor or a chaotic attractor (seeTable 1for a comparison between models). In the first case, all neurons in a brain area are assumed to fire irregularly, and the system is in a stable asynchronous state. In this case, oscillations - if they exist - manifest only transiently as resonances in the network response (Mattia and Del Giudice, 2002). This type of dynamics was considered in the resting-state models ofGhosh et al. (2008c),Deco and Jirsa (2012)andCabral et al. (2012a). On the other hand, if the neurons in a brain area are assumed to fire synchronously with rhythmic periodicity due to recurrent excitation and inhibition, then neural populations display self-sustained oscillations (Borgers and Kopell, 2003; Brunel, 2000; Brunel and Wang, 2003) and the dynamics is described by a limit-cycle attractor. In the resting-state models ofDeco et al. (2009)andCabral et al. (2011)the brain areas were considered to be in this regime. Finally, if we consider that, due to nonlinear interactions between neurons, the local network exhibits intrinsic instabilities where non-periodic intermittent oscillations occur then they instantiate a chaotic dynamics as inHoney et al. (2009). Importantly, all these regimes are plausible and supported by experimental evidence. However, they imply different scenarios for the emergence of slow BOLD signal fluctuations (Fig. 20). 5.2.3 Time delays and noise In addition to the dynamical regime at the node level, other factors such as time delays or noise may - or not - play a role resting-state activity. Time delays between brain areas are usually on the order of 10-100ms and arise principally from finite axonal transmission speed and from synaptic/dendritic processes. Although neglecting time delays significantly reduces the cost of numerical computations, it may not be permissible if the neural populations have an oscillatory dynamics and the time delays are in the same order of magnitude as the oscillation period (Cabral et al., 2011; Deco et al., 2009; Ghosh et al., 2008a). However, if a model considers that resting-state activity arises purely from slow (noise-induced) fluctuations around the stable asynchronous state, then time delays do not alter the slow dynamics and may be neglected (Cabral et al., 2012a; Deco and Jirsa, 2012). Still, in this case it is important to note that delays may reduce the stability of the network states (Jirsa, 2009; Jirsa and Ding, 2004) and therefore affect the dynamics at faster time-scales. Regarding the question of whether noise plays or not an essential role in resting-state dynamics, it depends again on the assumptions made in the model. Noise in the nervous system originates from various sources (e.g. finite size effects, spontaneous synaptic vesicle release, temperature-dependent Brownian motion of molecules, stochastic opening of ion channels, etc.) (Faisal et al., 2008). It is generally accepted that noise contributes to the variability observed experimentally in neural responses. However, it is still unclear whether noise plays a strategic role in the nervoussystem, influencing the dynamical characteristics of a system and not just its time course (Frank et al., 1999; Winterer et al., 1999). For example, a certain level of noise may enhance the detection and transmission of weak (periodic) signals by threshold-like systems (Benzi et al., 1981; Kosko and Mitaim, 2001). This process, termed stochastic resonance, is proposed byDeco et al. (2009)to drive the activation and deactivation of the resting-state spatiotemporal patterns. At the computational level, noise in the nervous system may be modelled in different ways. A common approach is to perturb the population dynamics of each cortical node using additive uncorrelated Gaussian white noise (Cabral et al., 2012a; Deco et al., 2009; Ghosh et al., 2008b). In more detailed spiking models, noise is due to finite size effect originated by uncorrelated background Poissonian spike trains from a finite number of spiking neurons (Deco and Jirsa, 2012). In the reduction of the detailed spiking model to a dynamic mean field model (Deco et al., 2013c), it is shown that this finite size noise can be reduced to additive uncorrelated Gaussian white noise. 5.2.4 Potentials and pitfalls of the different models The detailed conductance-based biophysical model (Honey et al., 2007, 2009) was the first model to compare the simulations results with empirical resting-state functional connectivity and is, so far, the one that obtains the highest correlation (r=0.7) between the model and empirical data. Importantly, neural activity and the BOLD signal were simulated for 998 ROIs and only subsequently reduced to 66 brain areas, which has not been done in other models to reduce computation costs. Structured BOLD signal fluctuations were related to the sporadic emergence of self-organizing synchronized patterns from the chaotic activity of neural populations. Notably, neither noise nor time-delays were considered in this model. Despite the good fit with empirical functional connectivity, the exact mechanism at the genesis of these self-organized synchronized patterns was not addressed. In a posterior work using the same model but with a synthetic cortico-thalamic connectivity matrix (Freyer et al., 2011), it was shown that the sporadic synchronized states were associated with spontaneous increases in the power of 10Hz oscillations and were due to bi-stability in the system. However, the relationship between the BOLD signal fluctuations found inHoney et al. (2009)and the faster cortical rhythms (as inFreyer et al., 2011) has not been explored. In a different approach, the FitzHugh-Nagumo units used in the model ofGhosh et al. (2008a,b)displayed damped oscillations (at 10Hz) when receiving sufficient input. For a specific coupling strength, the units operate at the instability border and any input fluctuations modulate the amplitude of 10Hz oscillations on a slow time scale (due to the damping time constant). The slowest modes of the neural activity were found to dominate in variance, generating slow BOLD signal fluctuations that correlated across bran areas. Contrary to the scenario fromHoney et al. (2009), this scenario proposes an important role of noise and time-delays in the system. In addition, it suggests a relationship between the BOLD signal and the alpha-band oscillations which are typically observed in spontaneous EEG data. However, the results have not been quantitatively compared with empirical data because simulations were performed with the connectivity from 1 hemisphere of the macaque brain (for which no fMRI functional connectivity matrix was available). Using Wilson-Cowan units at the node level,Deco et al. (2009)proposed another scenario, in which brain areas display self-sustained activity in the gamma-frequency band (40Hz) in the spontaneous state. Although the coupling strength, the time delays and noise also play an important role in shaping the resting-state activity as inGhosh et al. (2008a,b), the BOLD signal fluctuations have a totally different source. In this case, BOLD signal increases are due to the synchronization of neural activity within a given structural module. As this work focused on reproducing the 2 anti-correlated RSNs found inFox et al. (2005), brain areas were divided in 2 modules and parameters were tuned such that the 2 modules compete to synchronize (i.e. when one synchronizes the other desynchronizes), leading to anti-correlated BOLD signal fluctuations. Although these anti-correlations were later shown to originate from the global signal regression (Murphy et al., 2009; Weissenbacher et al., 2009), the biophysical synchronization mechanisms proposed in this work cannot be ruled out. Perhaps the simplest model of resting-state activity is the one proposed inCabral et al. (2011)using a generalized version of the Kuramoto model adapted to incorporate a realistic brain connectivity matrix and time-delays. In this case, brain areas were assumed to display self-sustained oscillations in the gamma-frequency range and the system was in a critical regime between incoherence and synchrony, where subsets of nodes temporarily synchronized but the whole network never fully synchronized. Despite its simplicity, the simulated activity displayed slow BOLD signal fluctuations with a correlation structure that fairly approximated the empirical resting-state functional connectivity. This model proposes that resting-state activity is driven by slow fluctuations in the synchrony degree of structural modules. Although the model was tested with noise, the latter did not play a fundamental role in the dynamics, because fluctuations at the critical point occurred due to intrinsic instabilities in the system. From a different perspective, the scenario proposed inCabral et al. (2012a)considers that the resting-state activity arises solely from the reverberation of noise in the large-scale structural network on a slow time-scale. In this case, brain areas are in an asynchronous spontaneous state and the noise fluctuations (leading to BOLD signal fluctuations) are structured by the coupling matrix. Although this model provides a good fit of the empirical BOLD functional connectivity at a low computational cost, it discards any relationship between the BOLD signal fluctuations and faster oscillatory signals in the alpha, beta or gamma band. Although electrophysiological studies of the resting-state activity point to a different direction, this scenario cannot be totally ruled out. Deco and Jirsa (2012)modelled each brain area as an attractor network of spiking neurons. Simulations with the model showed an optimal fit with empirical data when the system was in a stable low-firing state but very close to the bifurcation to a regime where multiple stable states co-exist. In this context, it was proposed that the resting-state activity results from an exploratory behaviour where several "ghost" attractors are intermittently visited during rest. However, due to the models' complexity, the mechanistic origin of resting-state activity was better investigated using the reduction of this detailed spiking model into a dynamic mean field model (Deco et al., 2013c). Using this reduction, it was found that resting-state activity in the model arises from noise propagation and dynamical slowing down of fluctuations in the anatomical network. Also in this model, the relationship with the faster oscillatory dynamics was not explored. 5.3 Outlook and future developments The most important take-home message is that, despite the intrinsic differences between models, the optimal fit with the empirical data is always found for a working point (defined by the model parameters) where the system operates at the edge of an instability, i.e. at the critical point of a bifurcation. Transitions at the critical point of the instability (which may be noise-driven or generated by intrinsic instabilities of the system) originate fluctuations in brain activity. In an anatomically constrained dynamical system like the brain, these fluctuations are spatially shaped by the underlying network structure, leading to an optimal reproduction of the functional connectivity maps observed in brain activity during rest. So far, existing models of spontaneous activity have focused essentially on reproducing findings from resting-state fMRI, such as fitting averaged BOLD functional connectivity matrices obtained through correlation-based approaches. Although these matrices provide information about how the BOLD signal is correlated between all brain areas, they implicitly assume temporal stationarity, i.e. that relationships are constant throughout the length of the recording. However, recent studies have shown that within-network functional connectivity fluctuates over time (Allen et al., 2012; Chang and Glover, 2010; de Pasquale et al., 2012; Deco et al., 2013b; Handwerker et al., 2012; Hutchison et al., 2013; Smith et al., 2012). These time-varying profiles of RSNs may relate to spontaneously shifting network states and therefore, it is crucial to consider the non-stationary connectivity dynamics in the future to further constrain and validate resting-state models (ongoing work in our lab).Senden et al. (2012)demonstrated that the neuroanatomical connectivity of the cortex allows for efficient multi-threading between RSNs using large-scale computational simulations. In addition, they found a strong correlation between the mean degree of RSNs and the proportion of time they are active. These results suggest that, even if it has not been adequately addressed, existing resting-state models display non-stationary connectivity dynamics. Another path to follow in further modelling studies is to explore the faster oscillatory dimension, revealed with other neuroimaging modalities, such as EEG, intracranial recordings and MEG (Cabral et al., 2013). Going beyond the BOLD signal, these methods provide pictures of resting-state dynamics from new perspectives,which must be taken into account in future models to deepen our understanding on resting-state activity. A common assumption made by the models, is that all brain areas are identical, and the global network activity is essentially determined by global parameters such as the global coupling weight, the global transmission speed or the global noise level. However, each brain area is different and the local parameters can be tuned to fit different aspects of their behaviour (i.e. power spectrum, variability of BOLD responses, etc.). As such, future extensions to resting-state models may local balance of the local dynamics to improve the performance of the models (work in progress in our lab). 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